In the grand canonical ensemble, the particle number *N* is
not constant. It is, therefore, instructive to calculate
the fluctuation in this quantity. As usual, this is defined to be

Note that

Thus,

In order to calculate this derivative, it is useful to introduce the Helmholtz free energy per particle defined as follows:

where is the volume per particle.

The chemical potential is defined by

Similarly, the pressure is given by

Also,

Therefore,

and

But recall the definition of the isothermal compressibility:

Thus,

and

and the relative fluctuation is given by

Therefore, in the thermodynamic limit, the particle number fluctuations vanish, and the grand canonical ensemble is equivalent to the canonical ensemble.

Tue Feb 1 14:50:00 EST 2000